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Theorem elmpps 32422
Description: Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mppsval.p 𝑃 = (mPreSt‘𝑇)
mppsval.j 𝐽 = (mPPSt‘𝑇)
mppsval.c 𝐶 = (mCls‘𝑇)
Assertion
Ref Expression
elmpps (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)))

Proof of Theorem elmpps
Dummy variables 𝑎 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ot 4483 . . 3 𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴
2 mppsval.p . . . 4 𝑃 = (mPreSt‘𝑇)
3 mppsval.j . . . 4 𝐽 = (mPPSt‘𝑇)
4 mppsval.c . . . 4 𝐶 = (mCls‘𝑇)
52, 3, 4mppsval 32421 . . 3 𝐽 = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
61, 5eleq12i 2874 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
7 oprabss 7119 . . . 4 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ ((V × V) × V)
87sseli 3887 . . 3 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
92mpstssv 32388 . . . . . 6 𝑃 ⊆ ((V × V) × V)
109sseli 3887 . . . . 5 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ((V × V) × V))
111, 10syl5eqelr 2887 . . . 4 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
1211adantr 481 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)) → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
13 opelxp 5482 . . . 4 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) ↔ (⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V))
14 opelxp 5482 . . . . 5 (⟨𝐷, 𝐻⟩ ∈ (V × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V))
15 simp1 1129 . . . . . . . . . 10 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → 𝑑 = 𝐷)
16 simp2 1130 . . . . . . . . . 10 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → = 𝐻)
17 simp3 1131 . . . . . . . . . 10 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → 𝑎 = 𝐴)
1815, 16, 17oteq123d 4727 . . . . . . . . 9 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → ⟨𝑑, , 𝑎⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
1918eleq1d 2866 . . . . . . . 8 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → (⟨𝑑, , 𝑎⟩ ∈ 𝑃 ↔ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃))
2015, 16oveq12d 7037 . . . . . . . . 9 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → (𝑑𝐶) = (𝐷𝐶𝐻))
2117, 20eleq12d 2876 . . . . . . . 8 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → (𝑎 ∈ (𝑑𝐶) ↔ 𝐴 ∈ (𝐷𝐶𝐻)))
2219, 21anbi12d 630 . . . . . . 7 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → ((⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)) ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
2322eloprabga 7120 . . . . . 6 ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
24233expa 1111 . . . . 5 (((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
2514, 24sylanb 581 . . . 4 ((⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
2613, 25sylbi 218 . . 3 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
278, 12, 26pm5.21nii 380 . 2 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)))
286, 27bitri 276 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2080  Vcvv 3436  cop 4480  cotp 4482   × cxp 5444  cfv 6228  (class class class)co 7019  {coprab 7020  mPreStcmpst 32322  mClscmcls 32326  mPPStcmpps 32327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-op 4481  df-ot 4483  df-uni 4748  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-iota 6192  df-fun 6230  df-fv 6236  df-ov 7022  df-oprab 7023  df-mpst 32342  df-mpps 32347
This theorem is referenced by:  mthmpps  32431  mclspps  32433
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